Stable Polynomials over Finite Fields

TL;DR

This paper investigates the stability of polynomials over finite fields by using resultants and character sums, extending previous quadratic cases and providing bounds on the number of stable polynomials.

Contribution

It generalizes the stability analysis from quadratic to arbitrary polynomials over finite fields and applies the Weil bound to estimate stable polynomial counts.

Findings

Partial generalization of quadratic polynomial stability to arbitrary polynomials

Identification of non-stability of certain degree three polynomials in characteristic three

Estimation of the number of stable polynomials using Weil bounds

Abstract

We use the theory of resultants of polynomials to study the stability of an arbitrary polynomial over a finite field, that is, the property of having all its iterates irreducible. This result partially generalises the quadratic polynomial case described by R. Jones and N. Boston. Moreover, for characteristic three, we show that certain polynomials of degree three are not stable. We also use the Weil bound for multiplicative character sums to estimate the number of stable arbitrary polynomials over finite fields of odd characteristic.